How is UN Security Council responsible for crime against humanity?
As an international governing body, the United Nations is seen as a "world umpire." Their purpose is to not necessarily favor any particular nation's agenda, but advance the cause for all of humanity, identify particular challenges that create struggle on a universal or multinational level, and propose/ implement methods designed to alleviate such challenges.
The United Nations consists of the General Assembly, where every nation receives a vote on essential issues, and the Security Council, whose decisions are to be followed by all UN Nations. The Security Council is able to enforce its decisions, as opposed to the General Assembly who can advocate and motion, but has no valid way of enforcing its decisions.
The five permanent members of the Security Council- United States, Russia, China, Britain, and France- each have the power to check one another and the Council must have a unanimous vote in order for the Council to act. It is in this element where the Security Council is responsible to act in crimes against Humanity. The idea would be that the members of the Security Council would have a shared understanding of the situation and would be able to unanimously act to solve a problem.
Their level of responsibility is high and valid, but at the same time, the ability to veto or reject a vote makes action possible only on a nearly universal level. Still decisions can be made if a nation abstains, or does not cast a vote. For example, the Security Council has voted on a resolution to condemn the genocide in Darfur by disarming the militia, but China, who has connection to some of the rebels in the Darfur militia with its supply of weapons and possession of oil, abstained from voting.